Tear film dynamics on an eye-shaped domain I: pressure boundary conditions

Maki, Kara L.; Braun, Richard; Henshaw, William D.; King‐Smith, P. Ewen

doi:10.1093/imammb/dqp023

Cited by 30 publications

(48 citation statements)

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“…The boundary was required to be so smooth to avoid introducing error from the boundary into the fluid flow. Details of the boundary shape can be found in our previous papers (Li et al, 2014b; Maki et al, 2010a, 2010b). We have used this shape for several kinds of mathematical models, including the new results in Section 5.3.4 of this paper.…”

Section: 1 Mathematical Model Parametersmentioning

confidence: 99%

Dynamics and function of the tear film in relation to the blink cycle

Braun

King‐Smith

Begley

et al. 2015

Progress in Retinal and Eye Research

118

163

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Great strides have recently been made in quantitative measurements of tear film thickness and thinning, mathematical modeling thereof and linking these to sensory perception. This paper summarizes recent progress in these areas and reports on new results. The complete blink cycle is used as a framework that attempts to unify the results that are currently available. Understanding of tear film dynamics is aided by combining information from different imaging methods, including fluorescence, retroillumination and a new high-speed stroboscopic imaging system developed for studying the tear film during the blink cycle. During the downstroke of the blink, lipid is compressed as a thick layer just under the upper lid which is often released as a narrow thick band of lipid at the beginning of the upstroke. “Rippling” of the tear film/air interface due to motion of the tear film over the corneal surface, somewhat like the flow of water in a shallow stream over a rocky streambed, was observed during lid motion and treated theoretically here. New mathematical predictions of tear film osmolarity over the exposed ocular surface and in tear breakup are presented; the latter is closely linked to new in vivo observations. Models include the effects of evaporation, osmotic flow through the cornea and conjunctiva, quenching of fluorescence, tangential flow of aqueous tears and diffusion of tear solutes and fluorescein. These and other combinations of experiment and theory increase our understanding of the fluid dynamics of the tear film and its potential impact on the ocular surface.

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Section: 1 Mathematical Model Parametersmentioning

confidence: 99%

Dynamics and function of the tear film in relation to the blink cycle

Braun

King‐Smith

Begley

et al. 2015

Progress in Retinal and Eye Research

118

163

View full text Add to dashboard Cite

show abstract

“…To our knowledge, Maki et al (2010a, b) were the first to extend models of fluid dynamics in the tear film to a geometry that approximated the exposed ocular surface. They formulated a relaxation model on a stationary 2D eye-shaped domain that was approximated from a digital photo of an eye.…”

Section: Introductionmentioning

confidence: 99%

Computed tear film and osmolarity dynamics on an eye-shaped domain

Li¹,

Braun²,

Driscoll³

et al. 2015

Math Med Biol

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The concentration of ions, or osmolarity, in the tear film is a key variable in understanding dry eye symptoms and disease. In this manuscript, we derive a mathematical model that couples osmolarity (treated as a single solute) and fluid dynamics within the tear film on a 2D eye-shaped domain. The model includes the physical effects of evaporation, surface tension, viscosity, ocular surface wettability, osmolarity, osmosis and tear fluid supply and drainage. The governing system of coupled non-linear partial differential equations is solved using the Overture computational framework, together with a hybrid time-stepping scheme, using a variable step backward differentiation formula and a Runge–Kutta–Chebyshev method that were added to the framework. The results of our numerical simulations provide new insight into the osmolarity distribution over the ocular surface during the interblink.

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“…First, our boundary conditions at x = ±L can be expressed in the form of the "pressure" boundary conditions considered by Bertozzi et al with p = 4 (14) 2 /9 ≈ 87 > 2. Second, our function F (h) in (52), defined specifically for the various models and boundary conditions by (53), (B4), (54), and (B9), has the form F (h) ∼ h n in the limit h → 0.…”

Section: Dimensionless Equationsmentioning

confidence: 99%

“…Without a sufficient tear film, eye irritation may occur and can lead to more severe damage of the corneal surface. The understanding of diseases such as dry eye syndrome has led to a growing interest in the applied mathematics and fluid dynamics community in developing models that allow for quantitative prediction of tear film thinning and rupture addressing issues such as aqueous layer stability (Sharma and Ruckenstein [8]), the dynamics of tear deposition and thinning due to lid motion (Wong, Fatt, and Radke [9]), mucus layer stability (Sharma, Khanna, and Reiter [1]), non-Newtonian rheology of the tear film (Zhang, Matar, and Craster [2]), evaporation and gravitational drainage (Braun and Fitt [3]), evaporation and corneal surface wetting (Winter, Anderson, and Braun [10]), dynamics during blink cycles (Heryudono et al [11], Braun and King-Smith [12]), reflex tearing (Maki et al [13]) and two-dimensional eye-shaped domains (Maki et al [14,15]). …”

mentioning

confidence: 99%

Thin Film Evolution over a Thin Porous Layer: Modeling a Tear Film on a Contact Lens

Nong¹,

Anderson²

2010

SIAM J. Appl. Math.

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Abstract.We examine a mathematical model describing the behavior of the precontact lens tear film of a human eye. Our work examines the effect of contact lens thickness and lens permeability on the film dynamics. Also investigated are gravitational effects and the effects of different slip models at the fluid-lens interface. A mathematical model for the evolution of the tear film is derived using a lubrication approximation applied to the hydrodynamic equations of motion in the fluid film and the porous layer. The model is a nonlinear fourth-order partial differential equation subject to boundary conditions and an initial condition for post-blink film evolution. The evolution equation is solved numerically, and the effects of various parameters on the rupture of the thin film are studied. We find that increasing the lens thickness, permeability, and slip all contribute to an increase in the film thinning rate, although for parameter values typical for contact lens wear, these modifications are minor. Gravity plays a role similar to that for tear films in the absence of a contact lens. The presence of the contact lens does, however, fundamentally change the nature of the rupture dynamics as the inclusion of the porous lens leads to rupture in finite time rather than infinite time. [5], and Cher [6]) suggest a somewhat more complex system. In particular, the current view replaces the mucus and aqueous layers with a mucoaqueous layer in which mucins secreted from goblet cells are distributed throughout the bulk of the tear film and epithelial mucins form a complex barrier at the corneal surface. Measurements of the overall thickness of a human tear film range from a few microns to as many as 40 (see the review by Bron et al. [5]). The corneal surface itself has 100-nm-scale microbumps and rod-like mucins of length 200-500 nm that extend into the tear film [4]. These mucins have multiple functions ranging from cleanup and removal of debris from the tear film, stabilization of the tear film, and, with particular attention to the membrane-associated mucins, maintaining wettability at the corneal surface. The lipid layer, whose thickness has been estimated to range between 13-100 nm, serves to further stabilize the film and to slow evaporative mass loss (Bron et al. [5]).Dry eye syndrome is a common disorder of the human tear film that results from decreased tear production, excessive tear evaporation, and/or an abnormality in the

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Tear film dynamics on an eye-shaped domain I: pressure boundary conditions

Cited by 30 publications

References 50 publications

Dynamics and function of the tear film in relation to the blink cycle

Dynamics and function of the tear film in relation to the blink cycle

Computed tear film and osmolarity dynamics on an eye-shaped domain

Thin Film Evolution over a Thin Porous Layer: Modeling a Tear Film on a Contact Lens

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